Phase-shifted Andreev levels in an altermagnet Josephson junction

We compute the effect of a d-wave magnetization (altermagnetism) on the spectrum of bound states (Andreev levels) in a junction between two s-wave superconductors (gap ${\mathrm{\Delta }}_0$, phase difference $\phi$). Compared to a nonmagnetic junction, the $\phi$-dependence of the spectrum is shifted by an offset $\pm \delta \phi$, dependent on the spin direction, so that the Andreev levels become spin-polarized. In a planar junction, oriented along the crystalline axis of $d_{xy}$-wave symmetry, the excitation energies are determined by the normal-state transmission probability $T$ according to $E={\mathrm{\Delta }}_0\sqrt{1-T{\sin }^2\frac{1}{2}(\phi \pm \delta \phi )}$. We calculate the corresponding Josephson energy and supercurrent, recovering the 0-$\pi$ transition of related studies.